Social Physics, Basic Laws in Social Complex Systems and Nonlinear Whole Sociology

International Journal of Modern Social Sciences, 2013, 2(1): 20-33 International Journal of Modern Social Sciences ISSN: 2169-9917 Journal homepage: www.ModernScientificPress.com/Journals/IJMSS.aspx Florida, USA Article Social Physics, Basic Laws in Social Complex Systems and Nonlinear Whole Sociology

Yi-Fang Chang

Department of Physics, Yunnan University, Kunming, 650091, China

E-Mail: [email protected]

Article history: Received 2 March 2013, Accepted 5 April 2013, Published 8 April 2013.

Abstract: We discuss generally the four variables and the eight aspects in social physics, and search social thermodynamics and the five fundamental laws of social complex systems. Then we research different relations among social elements, the moderate degree on the entropy production in systems and on the input negative entropy flow for open systems. Further, we discuss the evolutional equation of system and the educational equation, etc. Finally, we propose the nonlinear whole sociology and its four basic laws, and the nonlinear theory of economic growth.

Keywords: sociology, nonlinearity, social physics, thermodynamics, equation, law, economic growth

1. Introduction

Usual social physics (sociophysics) is the application of the concepts of physics in the social sciences. Some basic conceptions of physics, for example, force and energy, etc., were applied to society. Social physics is the term first coined by Auguste Comte to describe the synoptic vision of the unity of all science, social and physical. He discussed the social statics and the social dynamics. The modern physics has introduced many new theories and new methods, for instance, phase transition, chaos, dissipative structure, fractals, etc. They not only have mathematical characters, but also possess very high universality. Stewart proposed the suggested principles of social physics, and summarized the development of social physics (Stewart, 1947, 1950). T. M. Porter discussed a statistical survey of gases in Maxwell

Int. J. Modern Soc. Sci. 2013, 2(1): 20-33 21 social physics. E. S. Knowles studied social physics, social gravity models and the effects of audience size and distance, etc. S. P. Restivo researched the social relations of physics, mysticism and mathematics. D. A. Walker researched economics as social physics in The Economic Journal (1991). Bernard and Killworth searched social physics for social network knowledge and theory (Bernard, et al. 1997). Urry explored the increasing overlaps between sociology and physics, and explained the so- called small world phenomenon and corresponding new social physics (Urry, 2004). Warntz discussed transportation, social physics and the law of refraction (Warntz, 2005). B.Hillier discussed the relations between social physics and phenomenology for cities as large physical objects animated and driven by human behaviour. Glymour searched social science and social physics (Glymour, 2007). G.D.Snooks discussed self-organization or selfcreation from social physics to realist dynamics, which can be employed to explain and predict the emergence of social structures, even of history itself. Based on the synergetics (Haken, 1977, 1983), we proposed the social synergetics, and the four basic theorems, in which theorem of perfect correlation on humanity is researched mathematically (Chang, 2000a, 2013). The social synergetics is an application of synergetics, and both basic mathematics and equations are the same. From the synergetic equations, we obtained the equations on the rule of law, and proved mathematically that a society of the rule of law cannot lack any aspect for three types of the legislation, the administration and the judicature. Otherwise, we proposed an equation of corruption, and discuss quantitatively some threshold values for a social system into corruption. Further, from synergetics we obtain the Lorenz model, which may be a visualized two- party mechanism as a type of stable structure in democracy (Chang, 2013). A fundamental characteristic of social synergetics is a synergy with difference, even opposition, in which the synergy is a necessary condition to the existence of any social system. Determinacy and probability are complementary and unified in production and evolution. It is a harmony, or cooperation, or consistency passing reasonably consultation and balance in order to exist and develop for total social system, or to benefit majority. The social synergetics researches a general cooperation on different aspects and different levels. It discusses synergy from a world angle and from common benefit of humanity. A developed direction of society should be the combination from macroscopic to microscopic order, from an actual capable handling to an ideal pursuance (Chang, 2000a). Moreover, we discuss the synergetics on war and its basic synergetic principle, the slaving principle and the order-parameter combining some war-examples (Chang, 1999).

2. Social Physics and Its Eight Aspects

Statistics has widely applied in social science. Universally, any system can be classified into two types: an open system exchanging matter or energy or information with its surroundings, and an isolated system doing not (Chang, et al. 1987). Of course, both are distinguished relatively for different regions and levels. It is consistent with Bergson viewpoint and with the theory of dissipative structure. A complex social system is a set composed of many objects or elements with a certain relation, and this will have a high-dimensional space of many variables. By using the method of the modern physics, it may be projected on a low-dimensional space of few variables. Further, various main quantities are classified into: 1).The random variable  which cannot be dominated; 2).The relatively stable invariable in a certain region is called an extensive energy. 3).The extensive entropy which can describe various order degrees, organized powers, managed levels, and irreversibility, etc., and includes information. 4).The order parameter determining transitions of systems. Based on above analysis we can discuss the eight aspects except the social synergetics (Chang, 2000a, 2013): 1). The social kinetic energy K and the social potential energy U. 2). The social force F. It is defined by interaction between the social elements dU F  k . (1) dl Here l is an extension of the Bogardus-Simmel-Park social distance, or by C dK F  . (2) K dt Here C is the structure constant of a social system. 3). The social phase space. Its dimension equals independent of degree of freedom which describes essential character of a system. We consider that philosophy should be three dimensional for complex systems, so it may possess structure, stability, complexity and nonlinearity. 4). The social field. In the phase space a social field can be formed, whose function is

y  f (i , Ek ,S j ,t) . (3) For the average field

y  f (Ek ,S j ,t) . (4) The social field changes to pass through a critical threshold value, which may produce different structures of phase transition. This is a unification of necessity and fortuity. 5). The social quantum theory. In the society and history many quantities (country, man and so on) are quantized under distinguishable conditions of macroscopic parameter. It can be related with the extensive quantum theory (Chang, 1993, 2002, 2012b).

6). The social theory of dissipative structures (Chang et al. 1987). For a system, when it is open and the exchanged quantity reaches a threshold value, a new ordered structure can be formed. And if this quantity is provided sufficiently and constantly, such a stable self-organization can be maintained. 7). The social chaos. The nonlinear interactions among elements make usually system depended on initial conditions sensitively. When the order parameter reaches a certain vale, chaos appears, so economy collapses, war breaks out and so on. 8). The social fractals. It describes that the system possesses the self-similarity under some scaling transformations in a certain region. For example, some systems, the sizes of which are different, while whose structures are similar, and possess the same function. In this case the mathematics may apply the scale theory and the renormalization group.

3. Social Thermodynamics and Basic Laws on Social Complex Systems

The social thermodynamics is a part of the social physics (Galam, et al. 1982; Ball, 2002). Ilya Prigogine proposed order through fluctuation for self-organization and social system. Lepkowski discussed the social thermodynamics of Ilya Prigogine (Lepkowski, 1979). Reed and Harvey discussed complexity and realism in the social sciences, and critical philosophy and non-equilibrium thermodynamics (Reed, et al. 1992). J.L.R.Proops researched entropy, information and confusion in the social sciences. K.D.Bailey discussed social entropy theory and its application of nonequilibrium thermodynamics in human ecology, and living systems theory. Then he discussed living systems theory and social entropy theory (Bailey, 2006). G.A.Swanson, K.D.Bailey and J.G.Miller discussed social entropy and money in a living systems theory perspective. Balch researched hierarchic social entropy for an information theoretic measure of robot group diversity (Balch, 2000). E.L.Khalil studied nonlinear thermodynamics and social science modeling. Scafetta, et al., studied concretely the thermodynamics of social processes for the teen birth phenomenon. Zagreb researched an approach to a quantitative description of social systems based on thermodynamic formalism (Zagreb, 2000). Stepanic, et al., described social systems using social free energy and social entropy (Stepanic, et al. 2005). Statistical physics as the fruitful framework to describe phenomena, recent Castellano, et al., studied collective phenomena emerging from the interactions of individuals as elementary units in social structures of social dynamics, and emphasized a comparison of model results with empirical data from social systems (Castellano, et al. 2009). Further, Rifkin and Howard proposed entropy as a new world view (Rifkin, et al. 1981). Since a state of single element in any complex social and natural systems is indeterminate and fluctuated, we discussed the social thermodynamics, which analyse mainly that the total system agrees

Copyright © 2013 by Modern Scientific Press Company, Florida, USA Int. J. Modern Soc. Sci. 2013, 2(1): 20-33 24 with the statistical rules, and apply these methods which are analogous with thermodynamics and statistics. In the social thermodynamics the social temperature may be defined by T  cK (t) , where K is an average value of the social kinetic energy. While the social entropy may be defined by dS=dU/T. The present sociology has applied the statistical method, we research the universal statistical theory and principles in sociology. There are correspondingly social critical phenomena and phase transition. The threshold value at a critical point may pass through fluctuation to obtain a bifurcation point, which can produce different results of phase transition. It is a unification of necessity (a certain condition, threshold value) and fortuity (fluctuation, bifurcation point). Its mathematics may apply the catastrophe theory. For any social system obeying statistical rule, we proposed the five fundamental laws: The zeroth law: State fluctuation law. It originates from the dynamic background of system and the indeterminacy of state of single element, from this the fluctuation property in society and the social wave-motion are derived. The first law: Extensive energy E conservation law. The extensive energy corresponds to the internal energy. For instance, all of population, natural resources, fund, land, time and so on are fixed. Earth is only one. Some fundamental facts are neglected usually. In an isolated system dE/dt=0. The second law: Extensive entropy S changing law. S=klnW, where W is number of possible states of all element in this system. The extensive entropy is connected with the effective free energy. Usually it increase in an isolated system, and may decrease or increase in an open system. The third law: Threshold transition law. If the extensive entropy reaches a maximum value in an isolated system, or exchanged quantity reaches a threshold value in an open system, the order parameter will change suddenly, and then system will exhibit a phase transformation, and will form a new state. It will be able to be an ordered dissipative structure, or a chaos state, or a more disordered, or an ordered but dead state which is similar with crystal. This is a bifurcation point which may produce different results. The fluctuations have an enlargement effect for reach of the threshold value. The fourth law: Exclusion or inclusion law. The two different relations, exclusion or inclusion, exist among elements of various systems, both may be obvious or hidden forms. A synergetic relation can be formed only for inclusion elements. The causes of all conflicts and wars are exclusion of various benefits or powers. The five laws are, respectively, random factor which cannot be controlled exactly, objective conditions which cannot be changed arbitrarily, basis of potentialities which may be exploited fully, key problems which must closely attach importance, and objects of study which should be distinguished strictly.

4. Different Relations among Social Elements and Moderate Degree on Input Negative Entropy Flow

According to the different interrelations, which correspond to association, symbolic interactionism, conflict and so on in the modern sociology (Ritzer, et al. 2004), among elements in the system, we may determinate a set of relations: 1). When elements have different characters and statistics, there is exclusion or inclusion each other among elements. 2). Among elements there can be the competitiveness or the cooperativeness which is the same with P.Kropotkin viewpoint. Both correspond to Park four social processes, and to the common restraint or common promotion of the Five-Element in Chinese traditional culture. 3). The relations among elements may be linear or nonlinear, the latter can derive bifurcation and chaos, etc. Any linear function y=cx+b changes (i.e., differential) to derive dy / dt  c , which only is an equal change. The nonlinear relations can derive various complex evolutional patterns. 4). The interrelations may produce complementarity, intersection, structure and so on. These laws and relations form the whole world and its rules, and may be applied to discuss various social problems, for instance, development of a man self, protection of ecosystem, creativity- thinking, especially, some important doctrines in economics and management. It may apply the determinant-stochastic relations in historical events. The social thermodynamics is combined with the entropy, it will be obtained that the sustainable development should be connected with the moderate degree on the entropy production in systems and on the input negative entropy flow for open systems. We propose the evolutional equation of system, whose one dimensional nonlinear evolutional equation of society is: dS / dt  S m  S n  F(t) . (5) Its different characters and solutions correspond to above various relations. If the evolutional equation is combined with the entropy, it will be obtained that the sustainable development should be connected with moderate degree on the entropy production in systems and on the input negative entropy flow for open systems (Chang, 2000a). For a simple linear case, m=n=1 and a stochastic factor F(t)=0, so Eq.(5) becomes dS  Sdt  Sdt . (6) Let S is entropy of system, includes the entropy produce and positive entropy flow  for Sdt dSi dSe open systems, and is an input negative entropy flow  , so Sdt dSe

  dS  dSi  dSe  dSe . (7)

In the dissipative structure theory, the total change of entropy for an open nonequilibrium system is . (8) dS  dSi  dSe

  Here dSi  0 is the entropy production inside the system, and dSe  dSe  dSe is the entropy flow, which may be positive or negative. The total entropy is given as

  S  S0  dS  S0  dSi  dSe  dSe  0 , (9) which and the entropy production are always positive. The total entropy can decrease when input entropy flow is negative. Therefore, the maximum entropy is

  Smax  S0  dSi  dSe  S0  dSe  0 . (10) This defines a quantitative region of moderate degree on input negative entropy flow for any open system. Its absolute value is always greater than zero, but the total entropy can never become negative. In the general goal for input negative entropy flow is: (a) An existing order structure is kept, such that negative entropy flow equal entropy production, and the total entropy is invariance so that

S  S0 =constant. (b) It allows for internal entropy fluctuations, which imply the construction of a new

  order structure. In the second case, it is common for dSe  0 and dSe  dSi , so that the total entropy decrease and dS<0. Under the condition defined in equation (10), an input value of negative entropy flow can be neither excessively large nor small. Excessively small values prohibit the existence of a dissipative structure and do not achieve the threshold value for transformation to a new order structure. Conversely, if the excessively large values are beyond the sustained power of the system itself or the particular circumstances governing the system, it will break various stabilities. Therefore, the moderate degree on input negative entropy flow includes a control of open degree in system and a selection of input time. The input negative entropy flow is determined by the internal conditions of system and is restricted by the external circumstances. In either case, all living systems are very complex. Their entropies can be neither overly large nor small and the input negative entropy must have a period. For any open system, a rational combination between the input period and the input amount of negative entropy flow is guaranteed for either a stable structure or for the continual transformation to new ordered structures. These conditions are suitable for any living system. The moderate degree of input negative entropy flow is a universal scientific law. It is suitable for various natural and social systems, and human.

5. Evolutional Equation in Social System

Because human develops blindly, now a series of serious crises, for example, population explosion, environmental pollution, consumption of exhausted resources, unable regeneration ecology has dried up, has press on towards whole humanity. These like the terrific spirits. A nightmare on decay and elimination of some old nations has got entangled with our Earth Village. Since humanity has been faced with these crises, a sustainable development is proposed. All society begins to attach importance to a corresponding theory. Although some brilliant theories are proposed, for example, the system dynamics as a human opinion and corresponding mathematical base, theoretical level and philosophical intensions should be researched and developed continuously. We consider that humanity and our total natural circumstances are a huge common system. It includes above very much interacting elements. Synergetics, as a quantitative cooperation theory of different parts of a system, may be considered as a strategy to copy with complex systems. Therefore, it can be applied to the sustainable development. The synergetic equations on single-mode laser (Haken, 1977; Chang, 2000a) for F(t)=0 and   b*b can become the Lotka-Volterra equations, whose solution is a cycle model with period. This corresponds to a circulation of natural resources. In human ecology, the Lotka-Volterra equations may be a zero-solution. For example, when water has dried up in a region, humanity cannot live. When the chaotic motion occurs, the formerly stable system is destabilized. By using the slaving principle, the Lotka-Volterra equation can become the logistic equation: d / dt  aE  2 . (11) Its solution is: aE   . (12) 1 cexp(aEt) which can be a soliton of  -form. This corresponds to increasing limit in economics of natural resources, etc. Moreover, based on the master equations, Eq.(11) may also be derived (Haken, 1977; Chang, 2000a). Assume that the general nonlinear evolutional equations are: dx / dt  ax  bxy  cxz , (13) dy / dt  ey  fyz  gyx , (14) dz / dt  hz  kzx  lzy . (15) In various cases, dz/dt=0 corresponds to the symbiosis model in the population-dynamics, dx/dt=0 corresponds to the competition model, and dy/dt=0 corresponds to the cycle model and the predator- prey relation. The equations may derive various limit cycles. In the synergetics between humanity and nature (Chang, 2000b), the order parameters are the threshold value of reaching limit and the critical point of circulationable state. We research a theory on

Copyright © 2013 by Modern Scientific Press Company, Florida, USA Int. J. Modern Soc. Sci. 2013, 2(1): 20-33 28 the sustainable development, so the cooperativeness must be emphasized. The latter may also be applied to a modern mathematical representation on common restraint or promotion of Five-Element in the Chinese traditional philosophy. Both correspond to the two solutions: An increasing limit and a cycle model. They show the two stages of development: 1). The moderate degree and limit on increase. 2). A good circulation on development. For a society of system dynamics, its essential characters should be orderly and regular, and corresponds to a final ecological balance. They are also that the system dynamics must solve two necessarily fundamental problems. The three equations (13)-(15) correspond to the three elements on Sky-Earth-Man in Chinese traditional philosophy. Lao-tzu, a chinese ancient philosopher, said: Man models on land, land models on sky, sky models on the Dao (law), Dao models on nature. Lao-Zhuang philosophy thought: Man should be harmonious with environment and nature, both forms a suitable circle, finally it will achieve the highest goal of unifying humanity-nature. In this case the highest principle is not competition each other, and is a cooperation not only among various nations or countries, but among humanity and other plants and animals, humanity and total nature. Our outlets are enterprising new energy sources, extending new living space and useful regions for humanity, expending new technology, and form a great industrial circulation like an agricultural ecological village. A chinese traditional poem described vividly the circulation: Fallen flowers are not merciless and useless They transform manures, and enrich flowers. It is also an ideal state of Buddhism: without differences between life and death, and between me and world. When  =0 and m=1, Eq.(5) is the Langevin equation. For the stochastic factor F(t)=0, we obtain the education equation (Chang, 2000a): dE  a(t)E . (16) dt Its solution is: E  E(0)exp[ a(t)dt]. (17) This shows that the social benefit E is an exponential relation with the educative outlay a(t) devoted directly.

6. Nonlinear Whole Sociology and Its Four Laws

Humanity as an inseparable whole on Earth possesses common environment and benefit. It is a clear example in theorem of perfect correlation on humanity of the social synergetics (Chang, 2000a, 2013). Based on the inseparability and correlativity of the social systems, and analogy with the

Copyright © 2013 by Modern Scientific Press Company, Florida, USA Int. J. Modern Soc. Sci. 2013, 2(1): 20-33 29 nonlinear whole biology (Chang, 2001, 2012b), we propose the nonlinear whole sociology and the four basic laws: First law: The inseparability exists always among different organises, constructures, functions and levels within various social systems, which determinates to the globality of the social systems. Second law: Many main characteristics, for example, self-organization and self-adjustment of social systems are produced from some especial structures of complex subsystems. From this the interaction and nonlinearity exist necessarily. It includes fractal structures and chaos, etc. Third law: From a microscopic community, city, clime to a gigantic nation and country, various social systems of different levels possess the totality and nonlinearity. Their diversity and complexity originate from various different nonlinear interactions. Fourth law: A basic property of any social systems as an open system is that this system and its environment (for example, nature, geography, polity, culture, etc., and other social systems) must be a whole. It corresponds to a generalized metabolism. Usually environment is regarded as a boundary condition of the system, but it and the social systems have often various nonlinear relations. In modern and postmodern sociological theory (Ritzer, 1997; Ritzer, et al. 2004), systems theory, network theory, globalization theory are whole theory, while structural functionalism, neofunctionalism, conflict theory, structuralism, poststructuralism, existentialism and symbolic interactionism, etc., are inevitably nonlinear theories. The totality and the nonlinearity are two basic social characters. They are closely related. Because of complexity, the inseparability, and the correlativity of the social systems, their description must apply the nonlinear theory with the interaction terms. Reversibly, if there is not the totality, any society cannot be formed, single people is not a society. If there is not the nonlinear interaction, the system cannot form the social structure. Even the gregarious animal must also be a nonlinear whole society.

7. Nonlinear Theory of Economic Growth

The economic system is a particular social system, and should be whole and nonlinear. Using the similar formulas of the preference relation and the utility function, we propose the confidence relations and the corresponding influence functions that represent various interacting strengths of different families, cliques and systems of organization. Since they can affect products, profit, prices, and so on in an economic system, and are usually independent of economic results, therefore, the system can produce a multiply connected topological economics (Chang, 2012a). If the political economy is an economy chaperoned polity, it will produce consequentially a binary economy. When the changes of the product and the influence are independent one another, they may be a node or saddle point. When the influence function large enough achieves a certain threshold value, it will form

Copyright © 2013 by Modern Scientific Press Company, Florida, USA Int. J. Modern Soc. Sci. 2013, 2(1): 20-33 30 a wormhole with loss of capital. Various powers produce usually the economic wormhole and various corruptions. It is a mathematical application to economics. Based on the main characteristics of knowledge economy, the four theorems on the knowledge economic theory are proposed, and the production function and basic equations are expounded. Some possible directions of the development on the knowledge economy and a sustainable development theory of new economics are discussed (Chang, 2001, 2012a). In order to overcome a series of crises of the economic linear growth, based on Eq.(12) and the mathematical economics (Fuente, 2000), we think that the nonlinear evolution is a universal rule for economic growth, and propose the nonlinear theory of economic growth and its three laws: First law: Economic takeoff-growth-stagnancy law. Any output and corresponding economic development all must pass a general nonlinear evolutional process from takeoff to growth and stagnancy, no matter what for various merchandises or any country. This is unreasonable and impossible that anybody requests a persistent linear growth of economy. It corresponds to the maximum of the social effective throughput, and corresponds to a maximum   aE in Eq.(12) of developed limit when time t increase continuously. This is a stagnancy dates of economic growth. Second law: Social conservation and economic decay law. For any society, since the original throughput outmoded gradually, the social ageing, the saturated marketplace; contrarily, employment, laborage, welfare, operating costs and so on will increase continuously, and add the resources consumed, the wastes raised, the environmental largeness press and so on. Such the corresponding social effective throughput and the original economy develop to a certain extremum, and will descend inevitably. Third law: Economic growth mode transition and new developed period law. Further development of social economy must exploits new merchandise and market, and adjust output configuration, and reform technique, and train personnel, so that boost up the immanence ability of social development and the international competitiveness. At the same time, the social framework and various personnel must readjust combination, and the management level raises up to follow the social development and new talented persons, new equipments, new outputs, new techniques and new capital introduced. Such the society should reform continuously to achieve a higher seedtime. This is namely to search new economic growth point for microeconomics. It corresponds to a development of the paradigms in science. The three laws on economic growth may be represented by Figure 1, in which CA expresses the first law, AA’ expresses the second law, and AB expresses the third law. It should be a medium-time mode of economic growth, and is also three developed phases of social economy. Point A and dotted line are related with the limits to growth (Meadows, et al. 1974). The third law connects to “the quality

Copyright © 2013 by Modern Scientific Press Company, Florida, USA Int. J. Modern Soc. Sci. 2013, 2(1): 20-33 31 ladder” (Solow, 2000), which expresses a new period of development. The second law expresses a seasonal recession. They agree with Lotka-Volterra model in ecology, and are two different foregrounds of economic evolution, and are two-bifurcation phenomena of nonlinear system. An infinite clone of the same developed mode will derive a disorder competition, and finally reach necessarily to chaos and economic crisis. Therefore, the nonlinear chaos economics is possibly related with the crisis economics.

Fig.1: The three laws of economic evolution

Therefore, the social open is a necessary condition for economic further development, but it must add corresponding social reform as a sufficient condition of economic development. It may combine the theorem of transformation from energy to quality on social development (Chang, 2000a, 2013). Rivera-Batiz discussed the relation among democracy, governance and economic growth (Rivera-Batiz, 2002).

8. Conclusion

An important developing direction of modern social sciences is wide application of various mathematical methods. This should include the evolutions of social systems, whose open systems are related with the input negative entropy flow. The whole and the nonlinearity are two basic characters of any social system. For these questions we can perform some quantitative discussions and calculations. It overcomes the infantilism of the social physics in 18th century, and gives a conception of the modern physics on mankind society in a thought space, and set up a bridge between physics and social systems.

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